Nonuniversality and Finite Dissipation in Decaying Magnetohydrodynamic Turbulence

Abstract

A model equation for the Reynolds number dependence of the dimensionless dissipation rate in freely decaying homogeneous magnetohydrodynamic turbulence in the absence of a mean magnetic field is derived from the real-space energy balance equation, leading to Cε = Cε,∞ + C/R− + O(1/R 2 − )), where R− is a generalized Reynolds number. The constant Cε,∞ describes the total energy transfer flux. This flux depends on magnetic and cross helicities, because these affect the nonlinear transfer of energy, suggesting that the value of Cε,∞ is not universal. Direct numerical simulations were conducted on up to 2048 3 grid points, showing good agreement between data and the model. The model suggests that the magnitude of cosmological-scale magnetic fields is controlled by the values of the vector field correlations. The ideas introduced here can be used to derive similar model equations for other turbulent systems.PACS numbers: 47.65. 52.30.Cv, 47.27.Jv, 47.27.Gs Magnetohydrodynamic (MHD) turbulence is present in many areas of physics, ranging from industrial applications such as liquid metal technology to nuclear fusion and plasma physics, geo-, astro-and solar physics, and even cosmology. The numerous different MHD flow types that arise in different settings due to anisotropy, alignment, different values of the diffusivities, to name only a few, lead to the question of universality in MHD turbulence, which has been the subject of intensive research by many groups [1][2][3][4][5][6][7][8][9][10][11][12]. The behavior of the (dimensionless) dissipation rate is connected to this problem, in the sense that correlation (alignment) of the different vector fields could influence the energy transfer across the scales [2,13,14], and thus possibly the amount of energy that is eventually dissipated at the small scales.For neutral fluids it has been known for a long time that the dimensionless dissipation rate in forced and freely decaying homogeneous isotropic turbulence tends to a constant with increasing Reynolds number. The first evidence for this was reported by Batchelor [15] in 1953, while the experimental results reviewed by Sreenivasan in 1984 [16], and subsequent experimental and numerical work by many groups, established the now wellknown characteristic curve of the dimensionless dissipation rate against Reynolds number: see [17][18][19][20] and references therein. For statistically steady isotropic turbulence, the theoretical explanation of this curve was recently found to be connected to the energy balance equation for forced turbulent flows [19], where the asymptote describes the maximal inertial transfer flux in the limit of infinite Reynolds number.For freely decaying MHD, recent results suggest that the temporal maximum of the total dissipation tends to a constant value with increasing Reynolds number. The first evidence for this behavior in MHD was put forward in 2009 by Mininni and Pouquet [21] using results from direct numerical simulations (DNSs) of isotropic MHD turbulence. The temporal maximum of the total ...

Coarse-grained incompressible magnetohydrodynamics: analyzing the turbulent cascades

“…Those works showed that, in a sense, vorticity and magnetic flux, both being Lagrangian invariants, can also undergo a cascade. Recent numerical studies [41][42][43] have also presented evidence in support of the existence of a zeroth-law for the total energy in MHD turbulence. All major theories of MHD turbulence [44][45][46][47] implicitly rely on the existence of a dissipative anomaly.…”

Local approach to the study of energy transfers in incompressible magnetohydrodynamic turbulence

“…They provide a framework in which the main features of the flow such as the forcing, the kinematic and magnetic viscosities, or the initial and boundary conditions may be precisely controlled. As such, they have been extensively used for the study of MHD turbulence, under a wide range of conditions [67][68][69][70][71][72][73]. In the end, DNS give access to all components of the relevant fields on a grid.…”

Finite Dissipation in Anisotropic Magnetohydrodynamic Turbulence

“…A comparison with [18] revealed that the dimensionless dissipation rate saturates to a finite value but the level of saturation depends on the strength of initial cross-correlation. Recently, Linkmann et al [32,33] have performed a series of investigations for similar analysis in isotropic MHD. By fitting a model equation, an asymptotic value of dimensionless dissipation rate, C ǫ,∞ = 0.265 ± 0.013, was found for nonhelical decaying MHD with no mean field.…”